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 DẠNG TOÁN PHÂN TÍCH ĐA THỨC THÀNH NHÂN TỬ







































1A = - 6x + 9 - y^2 + x^2
1. Rewrite it in the form a^2−2ab+b^2, where a=3 and b=x.
  3^2−2(3)(x)+x^2−y^2
2. Use Square of Difference:(a−b)^2=a^2−2ab+b^2.
  (3−x)^2−y^2
3. Use Difference of Squares: a^2−b^2=(a+b)(a−b).
  (3−x+y)(3−x−y)

1B = 9x^2 — 16y^2
1. Find the Greatest Common Factor (GCF).
  GCF = 33
2. Factor out the GCF. (Write the GCF first. Then, in parentheses, divide each term by the GCF.)
  3(\Large{\frac{9x^2}{3}+\frac{−6y^2}{3}})
3. Simplify each term in parentheses.
  3(3x^2−2y^2)

2A = x^2 + 4x + 4 - y^2
1. Rewrite it in the form a^2+2ab+b^2, where a=x and b=2.
  x^2+2(x)(2)+2^2−y^2
2. Use Square of Sum: (a+b)^2=a^2+2ab+b^2.
  (x+2)^2−y^2
3. Use Difference of Squares: a2−b2=(a+b)(a−b).
  (x+2+y)(x+2−y)

2B = 25x^2 — 4y^2
1. Rewrite it in the form a^2−b^2, where a=5x and b=2y.
  (5x)^2−(2y)^2
2. Use Difference of Squares: a^2−b^2=(a+b)(a−b).
  (5x+2y)(5x−2y)

3A = x^2 — 5xy - 4x + 20y
1. Factor out common terms in the first two terms, then in the last two terms.
  x(x−5y)−4(x−5y)
2. Factor out the common term x−5y.
  (x−5y)(x−4)

3B = 4x^2 — 4x + 1 — y^2
L = 4x^2 — 4x + 1 — y%2
= (4x^2 — 4x + 1) — y^2
= (2x)^2 -2.2x.1 +1^2 - y^2   (có dạng a^2 -2ab + b^2 = (a – b)^2
= [(2x)^2 -2.2x.1 +1^2] - y^2
= (2x – 1)^2 – y^2     (có dạng a^2 – b^2 = (a + b)(a - b)
= [(2x -1)+(y)] [(2x -1)-(y)]
= (2x -1 + y)(2x -1 –y)

4A = x^2 - 2xy + y^2 – 49
1. Rewrite it in the form a^2−2ab+b^2, where a = x and b = y.
  x^2−2(x)(y)+y^2−49
2. Use Square of Difference: (a−b)^2=a^2−2ab+b^2.
  (x−y)^2−49
3. Rewrite it in the form a^2−b^2, where a=x−y and b=7.
  (x−y)^2−7^2
4. Use Difference of Squares: a^2−b^2=(a+b)(a−b).
  (x−y+7)(x−y−7)

4B = 16x^2 - 4y^2
1. Find the Greatest Common Factor (GCF).
  GCF = 4
2. Factor out the GCF. (Write the GCF first. Then, in parentheses, divide each term by the GCF.)
  4\Large{(\frac{16x^2}{4}+\frac{−4y^2}{4})}
3. Simplify each term in parentheses.
  4(4x^2−y^2)
4.Rewrite 4x^2−y^2 in the form a^2−b^2, where a = 2x and b = y.
  4((2x)^2−y^2)
5. Use Difference of Squares: a^2−b^2=(a+b)(a−b).
  4(2x+y)(2x−y)

5A = x^2 — 12x + 36 — 81y^2
1. Rewrite it in the form a^2−2ab+b^2, where a=x and b=6.
  2x^2−2(x)(6)+6^2−81y^2
2. Use Square of Difference: 2(a−b)^2=a^2−2ab+b^2.
  (x−6)^2−81y^2
3. Rewrite it in the form a^2−b^2, where a=x−6 and b=9y.
  (x−6)^2−(9y)^2
4. Use Difference of Squares: a^2−b^2=(a+b)(a−b).
  (x−6+9y)(x−6−9y)

5B = 9x^2 — y^2
1. Rewrite it in the form a^2−b^2, where a=3x and b=y.
  (3x)^2−y^2
2. Use Difference of Squares: a^2−b^2=(a+b)(a−b).
  (3x+y)(3x−y)

6A = x^2 — 10xy + 25y^2 – 16
1. Rewrite it in the form a^2−2ab+b^2, where a=x and b=5y.
  x^2−2(x)(5y)+(5y)^2−16
2. Use Square of Difference: (a−b)^2 = a^2 − 2ab + b^2.
  (x−5y)^2−16
3. Rewrite it in the form a^2 − b^2, where a = x − 5y and b = 4 .
  (x − 5y)^2 − 4^2
4. Use Difference of Squares: a^2 − b^2=(a + b)(a − b).
  (x − 5y + 4)(x − 5y − 4)

6B = 4x^2 - 36y^2
1. Find the Greatest Common Factor (GCF).
GCF = 4
2. Factor out the GCF. (Write the GCF first. Then, in parentheses, divide each term by the GCF.)
  4\Large{(\frac{4x^2}{4}+\frac{−36y^2}{4})}
3. Simplify each term in parentheses.
  4(x^2−9y^2)
4.Rewrite x2−9y2 in the form a2−b2, where a=x and b=3y.
  4(x^2−(3y)^2)
5. Use Difference of Squares: a^2−b^2=(a+b)(a−b).
  4(x+3y)(x−3y)

7A = x^2 — 2x + 1 — y^2
1. Rewrite it in the form a2−2ab+b2, where a=x and b=1.
  x^2−2(x)(1)+1^2−y^2
2. Use Square of Difference: (a−b)^2=a^2−2ab+b^2.
  (x−1)^2−y^2
3. Use Difference of Squares: a^2−b^2=(a+b)(a−b).
  (x−1+y)(x−1−y)

8A = 49x^2 — 9y^2
1. Rewrite it in the form a^2−b%2, where a=7x and b=3y.
  (7x)^2−(3y)^2
2. Use Difference of Squares: a^2−b^2=(a+b)(a−b).
  (7x+3y)(7x−3y)

8A = 3x(5x + y) + 25x^2 — y^2
  3x(5x + y) + (5x)^2 — y^2
  3x(5x + y) + (5x - y)(5x +y)
  (5x + y) [3x +(5x - y)]
  (5x + y) (3x +5x - y)
  (5x + y) (8x - y)

8B = 121x^2— 64y^2
1. Rewrite it in the form a^2 − b^2, where a = 11x and b = 8y.
  (11x)^2−(8y)^2
2. Use Difference of Squares: a^2 − b^2=(a + b)(a − b).
  (11x + 8y)(11x − 8y)

9A = x^2 - 8x + 16 – y2
1. Rewrite it in the form a^2 − 2ab + b^2, where a = x and b = 4.
    x^2 − 2(x)(4) + 4^2 − y^2
2. Use Square of Difference: (a − b)^2 = a^2 − 2ab + b^2.
  (x−4)^2 − y^2
3. Use Difference of Squares: a^2 − b^2 = (a + b)(a − b).
  (x – 4 + y)(x – 4 − y)

11A = x^2— 2x — y^2 + 1
= x^2 - 2x + 1 - y^2
  = (x^2 - 2x + 1) - y^2
  = (x – 1)^2 - y^2
  = (x – 1 + y)(x – 1 - y)

11B = 16x^2 — 144y^2
1. Find the Greatest Common Factor (GCF).
  GCF = 16
2. Factor out the GCF. (Write the GCF first. Then, in parentheses, divide each term by the GCF.)
  16(x^2) – 16(9y^2)
3. Simplify each term in parentheses.
  16(x^2 − 9y^2)
4.Rewrite x2 − 9y2 in the form a^2 – b^2, where a = x and b = 3y.
  16(x^2 − (3y)^2)
5. Use Difference of Squares: a^2 – b^2 = (a + b)(a − b).
  16(x + 3y)(x − 3y)

12A = x^2 — 6x + 9 — y^2
1. Rewrite it in the form a^2−2ab+b^2, where a = x and b = 3.
  x^2 − 2(x)(3) + 3^2 − y^2
2. Use Square of Difference: (a − b)^2 = a^2 − 2ab + b^2.
  (x−3)^2 − y^2
3. Use Difference of Squares: a^2 − b^2 = (a + b)(a − b).
  (x – 3 + y)(x – 3 − y)

12B = 81x^2 — 169ỵ^2
1. Rewrite it in the form a^2 − b^2, where a = 9x and b = 13y.
  (9x)^2 − (13y)^2
2. Use Difference of Squares: a^2 − b^2=(a + b)(a − b).
  (9x + 13y)(9x − 13y)

12C = xy — 3x + y^2 - 3y
  = (xy + y^2) — (3x + 3y)
  = y(x + y) — 3(x +y)
  = (x + y) (y— 3)

13A = x^2 + 10x + 25 — 64y^2
  = x^2 + 10x + 5^2 — 64y^2
  = (x^2 + 10x + 5^2) — 8^2y^2
  = (x + 5)^2 — (8y)^2
  = [(x + 5) + 8y] [(x + 5) - 8y]
  = (x + 5 + 8y) (x + 5 - 8y)

13B = x^2 — 16y^2
1. Rewrite it in the form a^2 − b^2, where a = x and b = 4y.
  x^2 − (4y)^2
2. Use Difference of Squares: a^2 − b^2 = (a + b)(a − b).
  (x + 4y)(x − 4y)

C = x^5 + x - 2x^4 – 2
1. Factor out common terms in the first two terms, then in the last two terms.
  x(x^4+1)−2(x^4+1)
2. Factor out the common term x^4+1.
  (x^4+1)(x−2)

14A = 5x(x — y) + 9x — 9y
  5x(x — y) + 9(x — y)
  = (x — y)(5x + 9)

14B = 196x^2 — 9y^2
  = (14x)^2 — (3y)^2
  = (14x)^2 — (3y)^2
  = (14x+3y)(14x−3y)

15A = x^2 — 4xy + 4y^2 — 36
1. Rewrite it in the form a^2 − 2ab + b^2, where a = x and b = 2y.
  x^2 − 2(x)(2y) + (2y)^2 − 36
2. Use Square of Difference:   (a − b)^2 = a^2 − 2ab + b^2.
  (x − 2y)^2 − 36
3. Rewrite it in the form a^2 − b^2, where a = x − 2y and b = 6.
  (x − 2y)^2 − 6^2
4. Use Difference of Squares:a^2 − b^2 = (a + b)(a − b).
  (x − 2y + 6)(x − 2y − 6)

15B = 225x^2 — 9y^2
1. Find the Greatest Common Factor (GCF).
  GCF = 9
2. Factor out the GCF. (Write the GCF first. Then, in parentheses, divide each term by the GCF.)
  9.25x^2 − 9.y^2)
  9\Large{(\frac{225x^2}{9} + \frac{−9y^2}{9})}
3. Simplify each term in parentheses.
  9(25x^2 − y^2)
4. Rewrite 25x^2 − y^2 in the form a2 − b2, where a = 5x and b = y.
  9((5x)^2 − y^2)
5. Use Difference of Squares:a^2 – b^2 = (a + b)(a − b).
  9(5x + y)(5x − y)

Bài 5a
Bài 55

Bài 6a
Bài 66

Bài 7a
Bài 77

Bài 8a
Bài 88

Bài 9a
Bài 99

Bài 10a
Bài 10

D = x^2 – 3
  a = x; b = \sqrt{3}
  -> E = x^2 – (\sqrt{3})^2
E có dạng a^2 – b^2 = (a +b)(a –b)E = (x + \sqrt{3})(x - \sqrt{3})

E = x^3 – 2
  2 = (\sqrt[3]{2})^3
E = x^3 - (\sqrt[3]{2})^3
  a = x; b = \sqrt[3]{2}
  a^3 – b^3 = (a – b)(a^2 + ab+ b^2)
E =(x - \sqrt[3]{2})(x^2 + \sqrt[3]{2} x+ (\sqrt[3]{2})2)
  (\sqrt[3]{2})^2 = 2^\frac{2}{3} =\sqrt[3]{2^2}
E = (x - \sqrt[3]{2}) (x^2 +\sqrt[3]{2} + \sqrt[3]{2^2} )

F = x^5 – 3x^3 – 2x^2 + 6
1. Factor out common terms in the first two terms, then in the last two terms.
  x^3(x^2−3)−2(x^2−3)
2. Factor out the common term x^2−3
  (x^2−3)(x^3−2)

G = x^2 - 2x + 1 - y^2
1. Rewrite it in the form a^2−2ab+b^2, where a=x and b=1.
  2x^2−2(x)(1)+1^2−y^2
2. Use Square of Difference: (a−b)^2=a^2−2ab+b^2.
  (x−1)^2−y^2
3. Use Difference of Squares: a^2−b^2=(a+b)(a−b).
  (x−1+y)(x−1−y)
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